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I have often heard it said that several problems in the theory of electromagnetism as described by Maxwell's equations led Einstein to his theory of Special Relativity. What exactly were these problems that Einstein had in mind, and how does Special Relativity solve them?

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You might want to be interested in reading mathpages.com/rr/s8-08/8-08.htm : ) –  Abel Molina Sep 5 at 3:05

4 Answers 4

There was no problem with electromagnetism. The problem was that Maxwell's equations are invariant under Lorentz transformations but are not invariant under Galileo transformations whereas the equations of classical mechanics can be easily made invariant under Galileo transformations.

The question was: how to reconcile both in a universe in which Maxwell's equations had been tested much more thoroughly than the equations of classical mechanics when $v$ is in the same order of $c$ and not much smaller.

Einstein basically solved the problem by deciding that electromagnetism is more fundamental in physics, and then showing that classical mechanics could be modified in such a way, that it, too, became Lorentz invariant. As a side effect, he recovered classical mechanics as a natural limit for $v/c\to0$, which perfectly explained almost all observations of macroscopic dynamics available at that time (leaving Mercury's perihelion precession to be explained by general relativity ten years later).

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This, I think, is the most illuminating way of looking at why SR works and was required. Thanks for that relatively (heh) fresh perspective, instead of the usual "traveling next to a light ray" stuff. –  Physics Llama Aug 31 at 22:38
    
Thank you, but the praise belongs to my high school physics teacher, who tried to lay these things out to me (and other students interested in material that went a bit beyond the classroom requirements) a long time ago. I hope that his neat explanation does not violate the actual history of relativity too much. –  CuriousOne Sep 1 at 5:03
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I would like to argue that Einstein did not have to decide that Lorentz boosts were more fundamental than Galilean boosts. The reason is that the changes that needed to be made to electromagnetism would contradict well-established experimental laws such as the Biot-Savart and Lorentz force laws. The changes required do not need large speeds to be observable! So well-tested experiments from 1905 and earlier were sufficient to rule out the possibility of having to modify the equations of electromagnetism. –  suresh Sep 2 at 0:08
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@suresh: The "controversy" was quite old in 1905. Maxwell's equations were around since 1862 and Lorentz transformations had been discussed at least since 1887. You are absolutely correct, that Einstein had all the pieces in his hand. What was missing, and what he supplied, was an authoritative verdict over the correct form of classical mechanics. Special relativity is therefor less of a discovery than it is a capping stone explanation put on the facts that were on the table for everyone to see. Einstein, however, saw them more clearly than others. –  CuriousOne Sep 2 at 2:23
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@CuriousOne I fully agree with what you say. My comment was meant to be a pedagogical one in that once one sees the conflict between Lorentz and Galilean boosts, nothing more needs to be done. Einstein was the first one to see it. –  suresh Sep 2 at 4:13

In at least one history I've read about Einstein's early life -- sorry, I don't recall the name of the book -- the author claimed that even back when Einstein was in Gymnasium (high school), he pondered a simple thought experiment:

What would an electromagnetic wave look like if one traveled along beside it at the speed of light?

The answer from Maxwell's own equations was, well... nothing! For example, if you visualize a plane polarized light front as alternating orthogonal potential lines of electric and magnetic potential that simultaneously generate each other and extinguish themselves through their forward motion, that forward generation process disappears from the view of an observer who is also moving at c. Without that motion, the very process by which the wave propagates and stays in existence ceases to exist. Strange!

According to that Einstein biographer, it was this rather simple conceptual thought problem that got Einstein enamored with the problem of objects traveling near or at the speed of light. Einstein realized something very important was missing from the setup, and he set about figuring out exactly what it was.

Einstein was deeply respectful of Maxwell, whom he referred to as one of the greatest physicists of all time. The respect is well justified, since Maxwell arguably came rather close to figuring out relativity decades before Einstein. I think he might well have done so had he not died so young.

Certainly Maxwell's equations did far more than just hint at relativity. Their implicit inclusion of invariance under the Lorentz transformation practically shouted the need for a new perspective, and in effect outlined the mathematical details of what that perspective would have to look like. It just took an innovative new way of looking at the implications, specifically Einstein's insight that any frame is as good as any other, to wrap up the package in its full generality as the special theory of relativity.

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Maxwell's equations of electromagnetism predicted that light would travel with a constant velocity c. The question is - a velocity c with respect to what? It was thus supposed that it must be with respect to an ether which was at absolute rest in the universe. It then followed from the Galilean transformation that absolute uniform motion with respect to the ether could be detected. But all attempts to detect such motion have failed. The most famous experiment being Michelson & Morley's interferometer.

This led Einstein to his first postulate in the theory of relativity: "Absolute uniform motion cannot be detected by any means". This is to say that the concept of absolute rest and the ether have no meaning. And the second postulate was that light is propagated in empty space with a velocity c which is independent of the motion of the source.

Einstein showed that in order for both postulates to be true we must modify our ideas about the nature of time.

A very nice example with a clock can be found in Feynman's lectures:

Suppose a simple clock built with two mirror pointing at each other (vertically), and a sensor which counts how many times the light bounced off of the mirrors: An observer in rest would see the distance between these mirrors as L, and the time each tick takes $\Delta t = \frac{L}{c}$. Now someone moving horizontally would see the path the light takes as $L^*=\sqrt{L^2+(v\Delta t^*)^2}$. So he would see the counter tick in $\Delta t^*=\frac{\sqrt{L^2+(v\Delta t^*)^2}}{c}$ , so ${\Delta {t^*}}^2(1-\frac{v^2}{c^2})=\frac{L^2}{c^2}=\Delta t^2$ Which leads to Einstein's equation for time dilation: $\Delta t^* = \frac{\Delta t}{\sqrt{1-\frac{v^2}{c^2}}}$ .

This allows the speed of light to be constant in all reference frames, and solves the problem of the ether.

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Even if the answers from CuriousOne, Terry Bollinger, Mr.WorshipMe are correct, the historical answer is not yet given. For instance, the invariance of the speed of light was not a problem, since this concept was not known before Einstein... who introduced it to define simultaneity !

As referred to the original Einstein's paper, the motivation for the introduction of the theory of special relativity was the so-called moving magnet and conductor problem. There is a wikipedia page about this "paradox" and the resolution Einstein gave to it. Also, the citation (and translation) of the introduction of the Einstein paper (on the electrodynamics of moving bodies) is given on the Wiki-page.

In short, suppose a magnet (supposed charge-free) moves relative to a conductor, which hosts charges. If you're in the magnet frame, the Lorentz force $F=E+v\times B$ can only has magnetic component, since the electric field is only present in the conductor, and the charges are moving with velocity $v$. In contrary, when you choose the frame of reference of the conductor, the charges are not moving and the Lorentz force is purely electric. So the "paradox" is: why does something electric in one frame is magnetic in an other ? The resolution of the "paradox" is the Faraday law, which connect time-dependent magnetic field and electric flux.

The way Einstein resolved this "paradox" is by promoting all frames of reference to be the same in space-time (whereas Newton/Galileo mechanics defined all frames of reference to be the same in space only), and defining simultaneity. This led him to find the Lorentz transform.

Note nevertheless that the Einstein's motivation is not really resolved in his paper (a point that is not well discussed even at the present days). Indeed, there is no clear way to define relativity for solid bodies (especially elastic), and then the moving magnet and conductor problem is not yet understood correctly, except for point-like magnet and conductor. Also, if the magnet and conductor have a mass, they most certainly move with a small speed, and then most certainly we should define electromagnetism at low speed...

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Despite Einstein's much later statements that he wasn't thinking about Michelson-Morley when he came up with SR, he still makes a reference to failures to identify a relative motion of Earth with regards to an ether in the original publication, which seems more than a bit strange. I would chose to trust the publication more than the later Einstein. –  CuriousOne Sep 10 at 9:27

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